Quadratic Functions Formula

Quadratic functions are a quadratic function is a polynomial function of degree 2, written as f(x) = ax^2 + bx + c with a!= 0, whose graph is a U-shaped.

The Formula

f(x)=ax2+bx+corf(x)=a(xโˆ’h)2+kf(x) = ax^2 + bx + c \quad \text{or} \quad f(x) = a(x-h)^2 + k

When to use: The path of a thrown ball โ€” rising then falling โ€” traces a parabola opening downward.

Quick Example

f(x)=x2โˆ’4x+3f(x) = x^2 - 4x + 3 โ€” a parabola opening up with vertex at (2,โˆ’1)(2, -1).

Notation

aa is the leading coefficient (determines opening direction), (h,k)(h, k) is the vertex, and x=โˆ’b2ax = -\frac{b}{2a} is the axis of symmetry.

What This Formula Means

A quadratic function is a polynomial function of degree 2, written as f(x)=ax2+bx+cf(x) = ax^2 + bx + c with aโ‰ 0a \neq 0, whose graph is a U-shaped curve called a parabola that opens upward when a>0a > 0 or downward when a<0a < 0.

The path of a thrown ball โ€” rising then falling โ€” traces a parabola opening downward.

Formal View

A quadratic function f:Rโ†’Rf: \mathbb{R} \to \mathbb{R} has the form f(x)=ax2+bx+cf(x) = ax^2 + bx + c with aโ‰ 0a \neq 0. Its zero set is {xโˆˆRโˆฃax2+bx+c=0}\{x \in \mathbb{R} \mid ax^2 + bx + c = 0\}, with โˆฃzerosโˆฃโˆˆ{0,1,2}|\text{zeros}| \in \{0, 1, 2\} determined by sgnโก(b2โˆ’4ac)\operatorname{sgn}(b^2 - 4ac).

Worked Examples

Example 1

easy
Find the vertex of f(x)=x2โˆ’6x+8f(x) = x^2 - 6x + 8.

Answer

(3,โˆ’1)(3, -1)

First step

1
The xx-coordinate of the vertex is x=โˆ’b2a=โˆ’โˆ’62(1)=3x = -\frac{b}{2a} = -\frac{-6}{2(1)} = 3.

Full solution

  1. 2
    The yy-coordinate is f(3)=9โˆ’18+8=โˆ’1f(3) = 9 - 18 + 8 = -1.
  2. 3
    The vertex is (3,โˆ’1)(3, -1).
For a quadratic f(x)=ax2+bx+cf(x) = ax^2 + bx + c, the vertex formula x=โˆ’b2ax = -\frac{b}{2a} gives the axis of symmetry. Plugging this xx back in gives the minimum (if a>0a > 0) or maximum (if a<0a < 0) value.

Example 2

medium
Does the parabola g(x)=โˆ’2x2+4x+1g(x) = -2x^2 + 4x + 1 open upward or downward? Find its maximum value.

Example 3

medium
Find the vertex and axis of symmetry of f(x)=2x2โˆ’8x+3f(x) = 2x^2 - 8x + 3.

Common Mistakes

  • Forgetting aa may be negative - then the parabola opens downward and has a maximum, not a minimum.
  • Reading the vertex from the standard form directly - use x=โˆ’b2ax=-\frac{b}{2a} or convert to a(xโˆ’h)2+ka(x-h)^2+k.
  • Expecting one x-value per y - a horizontal line crosses a parabola twice, so two inputs can share an output.

Why This Formula Matters

Quadratics are the first non-straight function students master, introducing vertex, axis of symmetry, and the idea that one output can come from two inputs. Recognizing the ax2ax^2 term tells you to expect a curve and a turning point, not a constant rate. Recognizing it by "Is the highest power of the variable exactly 2, so the graph curves into a parabola?" โ€” rather than by familiar numbers โ€” is what lets a student tell it apart from linear function and quadratic formula and exponential function in a mixed problem set.

Frequently Asked Questions

What is the Quadratic Functions formula?

A quadratic function is a polynomial function of degree 2, written as f(x)=ax2+bx+cf(x) = ax^2 + bx + c with aโ‰ 0a \neq 0, whose graph is a U-shaped curve called a parabola that opens upward when a>0a > 0 or downward when a<0a < 0.

How do you use the Quadratic Functions formula?

The path of a thrown ball โ€” rising then falling โ€” traces a parabola opening downward.

What do the symbols mean in the Quadratic Functions formula?

aa is the leading coefficient (determines opening direction), (h,k)(h, k) is the vertex, and x=โˆ’b2ax = -\frac{b}{2a} is the axis of symmetry.

Why is the Quadratic Functions formula important in Math?

Quadratics are the first non-straight function students master, introducing vertex, axis of symmetry, and the idea that one output can come from two inputs. Recognizing the ax2ax^2 term tells you to expect a curve and a turning point, not a constant rate. Recognizing it by "Is the highest power of the variable exactly 2, so the graph curves into a parabola?" โ€” rather than by familiar numbers โ€” is what lets a student tell it apart from linear function and quadratic formula and exponential function in a mixed problem set.

What do students get wrong about Quadratic Functions?

The procedure for quadratic functions is the easy part; the trap is forgetting aa may be negative. Asking "Is the highest power of the variable exactly 2, so the graph curves into a parabola?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Quadratic Functions formula?

Before studying the Quadratic Functions formula, you should understand: linear functions, exponents.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Quadratic Equations: Factoring, Completing the Square, and the Quadratic Formula โ†’
โ† Back to Quadratic Functions See All Examples Practice Problems